## Illinois Journal of Mathematics

### Remarks on the quantum Bohr compactification

Matthew Daws

#### Abstract

The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how Sołtan’s quantum Bohr compactification can be used to construct a “compactification” in this category. Depending on the viewpoint, different C$^{*}$-algebraic compact quantum groups are produced, but the underlying Hopf $*$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^{*}$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of Sołtan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^{\infty}(\mathbb{G})$, involving the antipode, which allows one to compute the Hopf $*$-algebra of the compactification of $\mathbb{G}$; we later study when the antipode assumption can be dropped. In the cocommutative case, we make a detailed study of Runde’s notion of a completely almost periodic functional—with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat{G}$.

#### Article information

Source
Illinois J. Math., Volume 57, Number 4 (2013), 1131-1171.

Dates
First available in Project Euclid: 1 December 2014

https://projecteuclid.org/euclid.ijm/1417442565

Digital Object Identifier
doi:10.1215/ijm/1417442565

Mathematical Reviews number (MathSciNet)
MR3285870

Zentralblatt MATH identifier
1305.43006

#### Citation

Daws, Matthew. Remarks on the quantum Bohr compactification. Illinois J. Math. 57 (2013), no. 4, 1131--1171. doi:10.1215/ijm/1417442565. https://projecteuclid.org/euclid.ijm/1417442565

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